How to control mesh curvature using an armature
You can use a few objects for this...

Here is the objects involved


Pick one of the circle meshes as the target mesh.

Use the other three remaining circular meshes as the "theoretical children".

Apply the Copy Scale Constraint to all three of these children.


Use these settings for the constraint on all three children.



Arrange the mesh objects as shown.



Here's a demo of the magic.




Appended The Working Blend File Using Drivers + Demo Animation of how to use it